Rational-Hull Certificates Become Independently Replayable for the First Time, No New Geometric Progress This Round: RHCERT-v0.1 Doubles Publication Candidates to 4/77

Round 34 (AMRAL-LUC-FC-R34, 2026-09-20) establishes the origin of the term "RHCert"/"RHCERT" that carries through into later documents in this research line, including Round 37: RHCert is short for Rational-Hull CERTificate, taken directly from this round's custom binary format's magic header, "L34RHC01" ("RHC" = Rational-Hull-Certificate), and from this round's own document wording; it is entirely about the Lebesgue Universal Covering Problem, and the letters "RH" are purely a two-letter coincidence overlapping with "Riemann Hypothesis" — with no other connection to it whatsoever. This is precisely the question that was already raised during this programme's internal review, and it is clarified explicitly and directly here. Mechanically, an RHCert is a compact binary file that, for every terminal leaf node of a given cell's witness tree, stores the exact rational-number vertex coordinates of the inscribed polygon used to prove that the leaf's area exceeds the target threshold — coordinates aligned to a fixed grid (integer X, Y, with x=X/10¹⁸, y=Y/10¹⁸), accompanied by a ".rhidx" random-access index file — so that a second, independent verifier can check the underlying geometric proof directly from the stored numbers, without rerunning or trusting the original search code (the verifier performs none of candidate generation, the SciPy hull computation, or the emitter's pass/fail determination — it accepts only certificate bytes). This round concretely implements the "RHCERT-v0.1" binary format, re-emitting two already-qualified cells — cell47 (2,163,069 vertices, 9228/9228 leaves PASS, minimum margin 6.75024723×10⁻⁹) and cell48 (2,109,700 vertices, 9285/9285 leaves PASS, minimum margin 1.03164178×10⁻⁷, with membership/convexity/exact-area failures at 0 for both trees) — in this new, byte-level replayable form; after delta-varint plus DEFLATE compression, each vertex costs about 3.5–3.9 bytes, so the complete rational certificate for a tree of about 9000 leaves needs only about 8 MiB, showing that full certificate persistence is practically feasible. Direct result: the number of cells at the programme's highest evidence tier, "publication-candidate," doubles from 2/77 to 4/77 ({47,48,69,70}); with cell68 also completing 9255/9255 mpmath/libmp lower PASS this round and passing both the marker68 independent upper bound and the MPFR upper-bound cross-check to be promoted to ARITHMETICALLY-CLOSED-PROTOTYPE, the broader "arithmetic-closed-or-better" tier reaches 5/77 ({47,48,68,69,70}) — this comes purely from upgrading the "evidence format" for already-established results, not from any new geometric search; this round explicitly states that no new global common-depth geometry wave occurred, and the global counts remain unchanged at geometry-complete 66/77 and 51/77 at common depth d=36. Research direction is chosen and led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol.